## Linear Operators: Spectral operators |

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Page 1332

Hence, by the remark following Corollary 1.5, the range of E(z) is at most n-

dimensional. This proves (b). To prove (c), we have only to show that the set of

have f ...

Hence, by the remark following Corollary 1.5, the range of E(z) is at most n-

dimensional. This proves (b). To prove (c), we have only to show that the set of

**eigenfunctions**of T is complete. Since for each element f of our Hilbert space wehave f ...

Page 1383

Again we are in the situation of Section 4, the interval being finite, the spectrum

being discrete, and the set of

conditions A and C, the unique solution of tao = Ao satisfying the boundary

condition ...

Again we are in the situation of Section 4, the interval being finite, the spectrum

being discrete, and the set of

**eigenfunctions**being complete. With boundaryconditions A and C, the unique solution of tao = Ao satisfying the boundary

condition ...

Page 1386

Thus, by Corollary 30 and the remarks following Theorem 16, the orthonormal

has already been noted; but the present derivation is of particular interest since it

...

Thus, by Corollary 30 and the remarks following Theorem 16, the orthonormal

**eigenfunction**associated with the eigenvalue 2 = –1/k” is 21/*k-1/*e”. This facthas already been noted; but the present derivation is of particular interest since it

...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

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adjoint extension adjoint operator algebra analytic B-algebra B-space Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients compact operator complex numbers constant continuous function converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality infinity integral interval kernel Lemma Let f Let Q linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real numbers satisfies sequence singular ſº solution spectral spectral theorem square-integrable subspace Suppose symmetric operator theory To(r To(t topology transform uniformly unique unitary vanishes vector zero